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C₂×#7154 #126647

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂⁴	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2);(3,4)(5,6)(7,8,9,11,10,12,13,14)(16,21)(17,25)(18,27)(19,26)(22,23)(24,30)(28,29);(4,6)(15,16,18,22,19,23,17,21,20,24,25,29,26,28,27,30)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#129099, C₂×C₈×#357, C₂×#128666
maximal normal subgroup of	C₂×#129099, C₂×C₈×#357, C₂×#128666
maximal quotient of	C₂×#94026, C₂×#94025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33843);